kirchhoff–love plate theory - wikipedia, the free encyclopedia
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Deformation of a thin plate highlightingthe displacement, the midsurface (red) andthe normal to the midsurface (blue)
Kirchhoff–Love plate theoryFrom Wikipedia, the free encyclopedia
The Kirchhoff–Love theory of plates is a twodimensional mathematical model that is used to determine thestresses and deformations in thin plates subjected to forces and moments. This theory is an extension of EulerBernoulli beam theory and was developed in 1888 by Love[1] using assumptions proposed by Kirchhoff. Thetheory assumes that a midsurface plane can be used to represent a threedimensional plate in twodimensionalform.
The following kinematic assumptions that are made in this theory:[2]
straight lines normal to the midsurface remain straight after deformationstraight lines normal to the midsurface remain normal to the midsurface after deformationthe thickness of the plate does not change during a deformation.
Contents
1 Assumed displacement field2 Quasistatic KirchhoffLove plates
2.1 Straindisplacement relations2.2 Equilibrium equations2.3 Boundary conditions2.4 Constitutive relations2.5 Small strains and moderate rotations
3 Isotropic quasistatic KirchhoffLove plates3.1 Pure bending3.2 Bending under transverse load3.3 Cylindrical bending
4 Dynamics of KirchhoffLove plates4.1 Governing equations4.2 Isotropic plates
5 References6 See also
Assumed displacement field
Let the position vector of a point in the undeformed plate be . Then
The vectors form a Cartesian basis with origin on the midsurface of the plate, and are the Cartesian coordinates on the midsurface of theundeformed plate, and is the coordinate for the thickness direction.
Let the displacement of a point in the plate be . Then
This displacement can be decomposed into a vector sum of the midsurface displacement and an outofplane displacement in the direction. We canwrite the inplane displacement of the midsurface as
Note that the index takes the values 1 and 2 but not 3.
Then the Kirchhoff hypothesis implies that
If are the angles of rotation of the normal to the midsurface, then in the KirchhoffLove theory
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Displacement of the midsurface (left) and of a normal (right)
Note that we can think of the expression for as the first order Taylor series expansion of the displacement around the midsurface.
Quasistatic KirchhoffLove plates
The original theory developed by Love was valid for infinitesimal strains and rotations. The theory was extended by von Kármán to situations wheremoderate rotations could be expected.
Straindisplacement relations
For the situation where the strains in the plate are infinitesimal and the rotations of the midsurface normals are less than 10° the straindisplacement relationsare
Using the kinematic assumptions we have
Therefore the only nonzero strains are in the inplane directions.
Equilibrium equations
The equilibrium equations for the plate can be derived from the principle of virtual work. For a thin plate under a quasistatic transverse load theseequations are
where the thickness of the plate is . In index notation,
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where are the stresses.
Bending moments and normal stresses Torques and shear stresses
Derivation of equilibrium equations for small rotationsFor the situation where the strains and rotations of the plate are small the virtual internal energy is given by
where the thickness of the plate is and the stress resultants and stress moment resultants are defined as
Integration by parts leads to
The symmetry of the stress tensor implies that . Hence,
Another integration by parts gives
For the case where there are no prescribed external forces, the principle of virtual work implies that . The equilibriumequations for the plate are then given by
If the plate is loaded by an external distributed load that is normal to the midsurface and directed in the positive direction,the external virtual work due to the load is
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The principle of virtual work then leads to the equilibrium equations
Boundary conditions
The boundary conditions that are needed to solve the equilibrium equations of plate theory can be obtained from the boundary terms in the principle of virtualwork. In the absence of external forces on the boundary, the boundary conditions are
Note that the quantity is an effective shear force.
Constitutive relations
The stressstrain relations for a linear elastic Kirchhoff plate are given by
Since and do not appear in the equilibrium equations it is implicitly assumed that these quantities do not have any effect on the momentum balanceand are neglected. The remaining stressstrain relations, in matrix form, can be written as
Then,
and
The extensional stiffnesses are the quantities
The bending stiffnesses (also called flexural rigidity) are the quantities
The KirchhoffLove constitutive assumptions lead to zero shear forces. As a result, the equilibrium equations for the plate have to be used to determine theshear forces in thin KirchhoffLove plates. For isotropic plates, these equations lead to
Alternatively, these shear forces can be expressed as
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where
Small strains and moderate rotations
If the rotations of the normals to the midsurface are in the range of 10 to 15 , the straindisplacement relations can be approximated as
Then the kinematic assumptions of KirchhoffLove theory lead to the classical plate theory with von Kármán strains
This theory is nonlinear because of the quadratic terms in the straindisplacement relations.
If the straindisplacement relations take the von Karman form, the equilibrium equations can be expressed as
Isotropic quasistatic KirchhoffLove plates
For an isotropic and homogeneous plate, the stressstrain relations are
The moments corresponding to these stresses are
In expanded form,
where for plates of thickness . Using the stressstrain relations for the plates, we can showthat the stresses and moments are related by
At the top of the plate where , the stresses are
Pure bending
For an isotropic and homogeneous plate under pure bending, the governing equations reduce to
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Here we have assumed that the inplane displacements do not vary with and . In index notation,
and in direct notation
The bending moments are given by
Derivation of equilibrium equations for pure bendingFor an isotropic, homogeneous plate under pure bending the governing equations are
and the stressstrain relations are
Then,
and
Differentiation gives
and
Plugging into the governing equations leads to
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Since the order of differentiation is irrelevant we have , , and . Hence
In direct tensor notation, the governing equation of the plate is
where we have assumed that the displacements are constant.
Bending under transverse load
If a distributed transverse load is applied to the plate, the governing equation is . Following the procedure shown in the previoussection we get[3]
In rectangular Cartesian coordinates, the governing equation is
and in cylindrical coordinates it takes the form
Solutions of this equation for various geometries and boundary conditions can be found in the article on bending of plates.
Derivation of equilibrium equations for transverse loadingFor a transversely loaded plate without axial deformations, the governing equation hasthe form
where is a distributed transverse load (per unit area). Substitution of the expressions forthe derivatives of into the governing equation gives
Noting that the bending stiffness is the quantity
we can write the governing equation in the form
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In cylindrical coordinates ,
For symmetrically loaded circular plates, , and we have
Cylindrical bending
Under certain loading conditions a flat plate can be bent into the shape of the surface of a cylinder. This type of bending is called cylindrical bending andrepresents the special situation where . In that case
and
and the governing equations become[3]
Dynamics of KirchhoffLove plates
The dynamic theory of thin plates determines the propagation of waves in the plates, and the study of standing waves and vibration modes.
Governing equations
The governing equations for the dynamics of a KirchhoffLove plate are
where, for a plate with density ,
and
Derivation of equations governing the dynamics of KirchhoffLove plates
The total kinetic energy of the plate is given by
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Therefore the variation in kinetic energy is
We use the following notation in the rest of this section.
Then
For a KirchhofLove plate
Hence,
Define, for constant through the thickness of the plate,
Then
Integrating by parts,
The variations and are zero at and . Hence, after switching the sequence of integration, we have
Integration by parts over the midsurface gives
Again, since the variations are zero at the beginning and the end of the time interval under consideration, we have
For the dynamic case, the variation in the internal energy is given by
Integration by parts and invoking zero variation at the boundary of the midsurface gives
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If there is an external distributed force acting normal to the surface of the plate, the virtual external work done is
From the principle of virtual work . Hence the governing balance equations for the plate are
Solutions of these equations for some special cases can be found in the article on vibrations of plates. The figures below show some vibrational modes of acircular plate.
mode k = 0, p = 1
mode k = 0, p = 2
mode k = 1, p = 2
Isotropic plates
The governing equations simplify considerably for isotropic and homogeneous plates for which the inplane deformations can be neglected. In that case weare left with one equation of the following form (in rectangular Cartesian coordinates):
where is the bending stiffness of the plate. For a uniform plate of thickness ,
In direct notation
For free vibrations, the governing equation becomes
Derivation of dynamic governing equations for isotropic KirchhoffLove plates
For an isotropic and homogeneous plate, the stressstrain relations are
where are the inplane strains. The straindisplacement relations for KirchhoffLove plates are
Therefore, the resultant moments corresponding to these stresses are
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The governing equation for an isotropic and homogeneous plate of uniform thickness in the absenceof inplane displacements is
Differentiation of the expressions for the moment resultants gives us
Plugging into the governing equations leads to
Since the order of differentiation is irrelevant we have . Hence
If the flexural stiffness of the plate is defined as
we have
For small deformations, we often neglect the spatial derivatives of the transverse acceleration of the plateand we are left with
Then, in direct tensor notation, the governing equation of the plate is
References1. A. E. H. Love, On the small free vibrations and deformations of elastic shells, Philosophical trans. of the Royal Society (London), 1888, Vol. série A, N° 17 p. 491–
549.2. Reddy, J. N., 2007, Theory and analysis of elastic plates and shells, CRC Press, Taylor and Francis.3. Timoshenko, S. and WoinowskyKrieger, S., (1959), Theory of plates and shells, McGrawHill New York.
See also
BendingBending of platesInfinitesimal strain theory
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Linear elasticityPlate theoryStress (mechanics)Stress resultantsVibration of plates
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